
Re: fom  01  preface
Posted:
Dec 9, 2012 4:06 AM


On 9 Dez., 03:06, fom <fomJ...@nyms.net> wrote:
> > A question: Do you believe that there are more than countably many > > finite words? > > Do you believe that you can use infinite words (not finite > > descriptions of infinite sequences). > > Do you believe that you can put in order what you cannot distinguish? > > There is a certain history here. > > As set theory developed, Cantor was confronted > with the notion of "absolute infinity".
He found it in the holy bible and with St. Augustin. > > I prefer to go with Kant: > > "Infinity is plurality without unity" > > and interpret the objects spoken of in typical > discussions of set theory as transfinite numbers.
Unfortunately that does not answer my question because these transfinite numbers belong to a countable set.
> But, I side with > Aristotle on the nature of what roles are played > by a deductive calculus.
Aristotle rejected actual infinity in mathematics. He said: Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it.
> Scientific demonstration > is distinct from dialectical argumentation that > argues from belief. In turn, that distinction > informs that a scientific language is built up > synthetically. The objects of that language > are individually described using definitions. > The objects of that language are individually > presumed to exist.
Yes, that's matheology: belief without proof.
Consequently, the > names which complete the "incomplete symbols" > exist as references only by virtue of the fact > that the first names introduced for use in the > science are a wellordered sequence.
This answers my question. > > Since I cannot possibly defend introducing > more than some finite number of names in > this fashion, the assumption of transfinite > numbers in set theory has a consequence. It > can be reconciled with this position only > if models of set theory are admissible as > such when they have a global wellordering.
Like the 10^140 monkeys in Indian mythology. As scientific and as useful.
Regards, WM

